A study of minimizing sequences.

*(English)*Zbl 0553.49017The authors consider the problem P: \(\inf\{f(x): x\in X\}\) where X is a subset of a normed space X and \(f: X\to {\mathbb{R}}\) is bounded from below on X and uniformly continuous on bounded sets. A sequence \(\{x_ i\}^{\infty}_{i=0}\subset X\) is called eventually feasible for P if \(\lim_{i\to \infty}\inf \{\| x_ i-x\| | x\in X\}=0\). A bounded eventually feasible sequence \(\{\hat x_ i\}\) is called locally minimizing for P if there exists a \(\rho >0\) such that for all eventually feasible sequences \(\{x_ i\}\) satisfying \(\limsup_{i\to \infty}\| \hat x_ i-x_ i\| \leq \rho\) we have \(\limsup_{i\in K, i\to \infty}f(\hat x_ i)\leq\limsup_{i\in K,i\to \infty}f(x_ i)\) for all infinite subsets \(K\subset \{0,1,2,...\}\). Next, the authors define \(\tilde X\) to be the class of all infinite sequences \(\{x_ i\}\subset X\), and the vector space \(X^ s\) to be \(\tilde X/\sim\) where \(\sim\) is the equivalence relation defined as follows: \(\{x_ i\}\sim \{y_ i\}\) if and only if \(\lim_{i\to \infty}\| x_ i-y_ i\| =0\). Now, let \(z\in X^ s\) and let \(\{x_ i\}\) be any sequence in the equivalence class z. The following definitions are independent of the particular choice of this sequence: \(X^ s=\{z\in X^ s| \limsup_{i\to \infty}\| x_ i\| <\infty\) and \(\{x_ i\}\) is eventually feasible}, \(f^ s(z)=\limsup_{i\to \infty}f(x_ i)\). A point \(\hat z\in X^ s\) is called a local minimizer for the problem \(P^ s:\) \(\min \{f^ s(z)| z\in X^ s\}\) if \(\{\hat x_ i\}\) is a locally minimizing sequence for P, where \(\{\hat x_ i\}\) is any sequence in the equivalence class \(\hat z.\) The authors present some (first and second order) necessary and sufficient conditions for \(\hat z\) to be a local minimizer for \(P^ s\). It is shown that, for the case where f is differentiable, the standard optimality conditions for P lead directly to corresponding optimality conditions for \(P^ s\). The case where f is merely Lipschitz continuous is also considered and a first order necessary optimality condition (stated in terms of Clarke generalized gradients) is obtained. Finally, the authors consider some specific optimization problems with equality and inequality constraints and formulate optimality conditions for minimizing sequences for these problems.

Reviewer: M.Studniarski